Hybrid Design of Non-Regular Linear Arrays With ... - IEEE Xplore

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 61, NO. 12, DECEMBER 2013

[9] S. G. Mao and Y. Z. Chueh, “Broadband composite right/left-handed coplanar waveguide power splitters with arbitrary phase responses and balun and antenna applications,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 243–250, 2006. [10] C. H. Tseng and C. L. Chang, “A broadband quadrature power splitter using metamaterial transmission line,” IEEE Microw. Wireless Comp. Lett., vol. 18, pp. 25–27, 2008. [11] P. S. Hall, “Design principles of sequentially fed, wide bandwidth, circularly polarized microstrip antennas,” Proc. Inst. Elect. Eng., vol. 136, pt. H, pp. 381–389, Oct. 1989. [12] U. R. Kraft, “An experimental study on 2 2 sequential rotation arrays with circularly polarized microstrip radiators,” IEEE Trans. Antennas and Propag., vol. 45, pp. 1459–1466, Apr. 1997. [13] K. L. Chung, “On the design of sequential-rotation arrays composed of CP-EMCP elements,” in Proc. Int. Conf. Antennas and Propagation. (ISAP), Singapore, 2006, pp. 1–5. [14] K. L. Chung and A. S. Mohan, “A circularly polarized stacked electromagnetically coupled patch antenna,” IEEE Trans. Antennas Propag., vol. 52, no. 5, pp. 1365–1369, 2004. [15] Y. X. Guo, K. W. Khoo, and L. C. Ong, “Wideband circularly-polarized planar antenna using broadband baluns,” IEEE Trans. Antennas Propag., vol. 56, no. 2, pp. 319–326, 2008. [16] S. L. S. Yang, R. Chair, A. A. Kishk, K. F. Lee, and K. M. Luk, “Study on sequential feeding networks for subarrays of circularly polarized elliptical dielectric resonator antenna,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 321–333, 2007. [17] K. L. Chung, “A wideband circularly polarized H-shaped patch antenna,” IEEE Trans. Antennas Propag., vol. 58, no. 10, pp. 3379–3383, 2010. [18] HFSS ver. 11 ANSYS Inc. [19] Murata Catalogues, GRM188 Chip Capacitors and LQG18 Chip Inductor [Online]. Available: www.murata.com

Hybrid Design of Non-Regular Linear Arrays With Accurate Control of the Pattern Sidelobes D. Sartori, G. Oliveri, L. Manica, and A. Massa Abstract—The hybridization of an analytical thinning technique based on almost difference sets (ADSs) and a convex programming (CP) strategy is proposed for designing non-regular linear arrays over a half-wavelength lattice that fit arbitrary upper bounds on the far-field pattern. The proposed approach combines the efficiency of the ADS procedure in defining a low-sidelobe thinned arrangement with the effectiveness of the CP approach in deducing the optimal tapering to control the beam shape. A set of numerical results is presented to assess features and potentialities of the ADS-CP method. Index Terms—Array antennas, linear arrays, thinned arrays, sidelobe level control, almost difference sets, convex programming.

I. INTRODUCTION Light, inexpensive, low-consumption, and low-complexity architectures comprising fewer elements than traditional fully-populated Manuscript received October 22, 2012; revised September 18, 2013; accepted September 22, 2013. Date of publication September 26, 2013; date of current version November 25, 2013. The authors are with the ELEDIA Research Center @ DISI, University of Trento, Povo I-38123 Trento, Italy (e-mail: [email protected]; [email protected]; [email protected]; andrea.massa@ing. unitn.it). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2013.2283602

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phased arrays are gaining more and more popularity in several practical scenarios including MIMO radars [1], imaging [2], and satellite [3] applications. Unfortunately, the design of non-uniform massively-sparse architectures complying with user-defined pattern constraints is a much more challenging task than using traditional uniform fully-populated layouts because of the reduced degrees-of-freedom (DOFs) [4]. To solve such a problem, several approaches have been proposed in the literature which either handle sparse (i.e., antennas arbitrarily placed within the aperture) [5]–[9] or thinned (i.e., element locations picked from a regular lattice) arrays [4]. While the former architectures usually yield improved performances in terms of sidelobe control, pattern shape properties, and sparsity factor [10], and they can also effectively deal with wideband signals [8], [9], they usually yield to cumbersome real-valued synthesis problems, especially if wide apertures are of interest [7]. Conversely, the latter are often preferred for the easier fabrication and the arising modularity, as well as for the possibility to efficiently formulate and solve the synthesis problems as binary ones (although they do not usually enable to fully control the pattern features) [4]. As a consequence, the design of thinned layouts has received a lot of attention especially for cost-constrained applications as proven by the large set of thinning methods proposed in the last fifty years including random approaches [12], dynamic programming [11], stochastic optimization [4], [10], and, more recently, analytical [13] and hybrid techniques [14]. In this framework, state-of-the-art thinning methodologies are often formulated in terms of a sidelobe level (SLL) minimization problem, which is solved by suitably displacing the uniformly-weighted elements over the reference lattice [4], [13]. Although such a choice guarantees the low complexity of the resulting architecture [4], it also enables few DOFs for the design procedure, thus often preventing a full control of the beam shape [13]. This is also true for analytical methodologies, such as those based on almost difference sets (ADSs) [13], which a-priori estimate the SLL, but do not satisfy by definition a user-defined far-field mask. To overcome this drawback, the synthesis of non-regular arrays that fulfill power-pattern constraints has been successfully addressed in [5] through a hybrid methodology that combines in a nested way a local search strategy based on convex programming (CP) with a simulatedannealing (SA) optimization tool. However, such an approach does not enable the specification of a target lattice to comply with [5] because of the use of the SA strategy to define the element locations within the aperture. Moreover, global optimization algorithms (such as the SA) require a an increasing number of functional evaluations to converge when the arrangement width (and thus the number of unknowns) enlarges [5], [4], [14]. Alternatively, the hybridization of an efficient thinning strategy with a local optimization tool is proposed in this work as an efficient complement to existing strategies aimed at synthesizing non-regular linear arrays over a equally-spaced lattice that fit arbitrary power pattern masks. a fully-analytMore specifically, the proposed approach comprises ical ADS-based thinning step [13] for deducing the geometrical displacement of the array elements within the prescribed position grid in a computationally-efficient fashion and the successive application of a CP solver [5] to compute the element excitations complying with user-defined pattern bounds. Such choices are motivated by the following considerations: • Efficiency—ADS-based thinned arrangements represent good starting points for hybrid aperiodic array design methodologies [14] because of the computational efficiency whatever the aperture size [13] and the generation of pattern with low and predictable sidelobes even when no tapering is applied [13];

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• Optimality—Once the element locations are set, the problem of finding the excitations whose radiated field complies with arbitrary upper bounds turns out to be convex [15]. Therefore, it can be optimally solved by means of a CP strategy [5], [15], [16].

Afterwards, the optimal excitations for each ADS layout are computed by substituting (3) in (1) to reformulate the array design problem as follows

II. PROBLEM STATEMENT AND HYBRID ADS-CP SOLUTION The problem of designing a thinned linear array comprising elements which complies with user-defined upper bounds on the power pattern can be formulated as follows: Array

Design

Problem:

subject to

Find

is maximized, subject to

(4)

and such that for all (5)

where and are the -th antenna position expressed as an integer ( being multiple of the lattice spacing in wavelengths the design frequency) and its excitation, rethe speed of light, and spectively, is the linear array aperture (still in ), (1) is the steering direction, is the array factor, is the user-defined mask function, are the diis the mainrections where to enforce the pattern constraint, and lobe width [5]. It is worth noticing that such a formulation enables any steering direction (e.g., including end-fire) and sidelobe profile to be specified by the designer. To solve the Array Design Problem, a hybrid procedure is proposed where the element locations over the aperture lattice are selected according to the ADS strategy [13], while the corresponding excitations are computed by means of a CP procedure [5]. More specifically, fol-ADS1 lowing the guidelines in [13], an (2) is firstly chosen (e.g., picked from [17] or built according to [18]) such (to guarantee that target aperture is not exceeded that [13]). A set of ADS layouts is then obtained as follows

(3) where is the cyclic shift used to generate the -th arrangement from the “seed” sequence [13]. For instance, the (10, 5, 2, 7)-ADS (taken from [17]) can be used to generate ADS arrays with elements displaced as follows

.. .

1If

is an ADS, the multiset has non-zero elements which repeat times, while the renon-zero elements repeat times. maining

Since the only unknowns in (4)–(5) are the excitation coefficients, this (sub)problem turns out to be equivalent to a CP minimization one [15], which can be solved by any operations research method able to compute the global minimum of a constrained minimization problem through a deterministic/local search.2 In this framework, both general purpose [19], [5], [20] and ad-hoc algorithms [15] have been considered. A procedure exploiting the Matlab subroutine ‘fmincon’ is here adopted because of its effectiveness and robustness [5], [20] as also proved in several other array design problems [16]. and have been computed for each , Once the final design (i.e., ) is selected trade-offs. For inaccording to a user-defined criterion among the stance, the following choice will be made hereinafter (6) where (7)

is the sidelobe level of the -th solution, which corresponds to the selection of the layout with the best SLL as done in several state-of-the-art thinning strategies [4], [13]. It is also worth pointing out that other criteria (e.g., minimum dynamic range , minimum ratio peak-to-average power ratio, or maximum aperture efficiency ) could be considered, as well. The overall hybrid ADS-CP procedure can be summarized as follows: 1) Problem Definition—Define the array aperture , the lattice , the spacing , the element number , the mainlobe size , and the angular grid . sidelobe mask and ; Set 2) Initialization—Select from [17] or build [18] a suitable -ADS such that ; according to (3); 3) Geometry Computation—Compute 4) Excitation Computation—Compute by applying the subroutine fmincon to solve (4)–(5); —Compute according to (7). If 5) Update , then update ; 2Because of the convex nature of the problem, the use of stochastic/global optimization strategies, although possible, is not advisable in this case because of their higher computational costs (caused by the number of cost function evaluations needed to converge) [4], [18].


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6) Convergence Check—If , then return and , else increase the shift index and go to Step 3. guarantees that the final Let us notice that the above procedure turns out highly layout complies with the user-defined lattice (3), efficient even for large arrangements since no global optimization is can allow the indirect control of the array amplitude performed, ratio (i.e., DRR) through the definition of a suitable selection criterion only CP-based local searches are needed. in Step 5, and III. NUMERICAL VALIDATION In this Section, a set of representative examples is discussed to assess the performance and features of the proposed hybrid methodology when dealing with different aperture sizes, element number, and pattern eleconstraints. More specifically, arrays comprising ments have been synthesized assuming both equi-ripple (Section III.A) as well as shaped (Section III.B) sidelobe profiles. In the following, has unless otherwise stated, an half-wavelength lattice has been set to the first null of the corresponding been assumed and ADS pattern. Moreover, the pattern mask has been uniformly-sampled (e.g., ) and within the visible range by setting choosing (8) following the guidelines in [5], [16]. A. Equi-Ripple Sidelobes The first numerical test deals with the synthesis of a arrangement over an aperture of width with an average that complies with the equi-ripple spacing dB, .3 By applying the mask (Table I) is firstly proposed procedure, the (96, 48, 23, 24)-ADS is computed chosen from [17] and the basic arrangement according to (3) [Fig. 1(a)]. The CP strategy is then used to determine by solving (4)–(5). The power pattern in Fig. 1(b) is , is computed by then generated whose figure of merit, means of (7) [Fig. 1(c)]. The process is then repeated until and the final layout [ —Fig. 2(b)] is selected according to (6) as indicated in Fig. 1(c). As expected, the hybrid procedure enables a more accurate SLL control [Fig. 2(a)] than that from the ADS strategy (Table I) thanks to the non-uniform weighting [Fig. 2(b)]. Moreover, Fig. 1(c) points out the importance of the cyclic-shift ADS-CP search to identify the optimal arrangement since the layout does not exactly meet the mask requirements [see inset of in Fig. 1(b)]. On the other hand, the optimal solution terms of SLL turns out to be optimal also for the DRR [Fig. 1(c)]. Generally speaking, different trade-off layouts in the SLL-DRR plane are usually yielded through the ADS-CP as for the synthesis of an arrangement with elements [ dB ], whose outcomes are reported in Fig. 3 starting . As it can be observed [Fig. 3], from the (172, 86, 42, 43)-ADS the sidelobes are suitably controlled despite the wider aperture and —Table I]. Such a result underlines the massive thinning [ the capability of the approach to enhance the features of thinned ardB vs. dB; rangements ( vs. —Table I), while dB—Table I]. giving DRRs of about 10 dB [ 3Thanks to the symmetries in , real-valued excitations could be assumed in the synthesis. However, for a fair evaluation of the technique which is formu), such feature of lated in general terms (i.e., will not be exploited hereinafter to simplify the design.

Fig. 1. Equi-Ripple Sidelobes . ADS arrangement, (a). Power pattern of the ADS and the ADS-CP seed layouts (b). Beand versus the cyclic shift index, , of havior of the ADS-CP layouts (c).

To further assess previous outcomes, other test cases have been ad, while keeping dressed by varying in the range (Fig. 4). Towards this end, the ADSs (Table I) [17] have been used as seed sequences for facing the synthesis of equi-ripple pat) are reported in Table I. As it terns whose descriptors (i.e., SLL and can be noticed, the ADS-CP provides an enhancement of the sidelobe control as the arrangement enlarges (e.g., dB vs. dB—Fig. 4). Moreover, such an improvement with respect to the isophoric ADS arrays turns out to be more evident in correspondence with wider apertures (e.g., dB vs. dB when —Fig. 4). This is actually indicated from the theoretical viewpoint, since the first sidelobe of ADS layouts is expected to have an asymptotic value equal dB as owing to the ADS properties [13], while the to ADS-CP method always gains in terms of available DoFs as the aperture increases. As concerns the bandwidth of the ADS-CP arrangements, the ( behavior of the SLL versus the normalized frequency being the working frequency) for the optimized layout —Table I) indicates that the sidelobes are well controlled in ( a frequency band of about one octave above the nominal design —Fig. 5), despite the ADS-CP frequency (i.e., method itself is actually formulated for narrowband syntheses (see Section II). This feature, which is confirmed also for wider arrange—Fig. 5), is motivated by the properties ments (e.g., of the ADS seed sequences, whose sidelobes are well controlled until is approximately equal to one wavelength at the working frequency [13]. For completeness, the DRR of the optimal layouts (Table I) indicates dB when , the DRR increases as that, while

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TABLE I EQUI-RIPPLE SIDELOBES. ADSS SEQUENCES [17] AND SYNTHESIS CONSTRAINTS AND RESULTS

Fig. 4. Equi-Ripple Sidelobes the array size .

Fig. 2. Equi-Ripple Sidelobes timal ADS-CP layout layout (b).

versus

. Power pattern of the op(a) and amplitudes of the corresponding Fig. 5. Equi-Ripple Sidelobes . normalized frequency

Fig. 3. Equi-Ripple Sidelobes and ADS-CP layouts.

. Behavior of SLL and

. Behavior of versus the cyclic shift index, , of the

wider apertures are at hand. This is mainly due to the constraint in (6) . As a matter of fact, (6) totally neglects any informafor selecting tion on the DRR of the cyclically shifted sequences. Therefore, better DRR values could be obtained also for larger arrangements with a different definition of (6) that forces the best SLL-DRR trade-off solution. On the other hand, the aperture efficiency of ADS-CP optimal arrangements turns out always above that of the corvs. responding ADS layout (e.g.,

. Behavior of SLL versus the

—Table I). This shows that the ADS-CP indirectly enhances also the aperture efficiency of the designed array [see (6)]. despite is not considered in the selection of As for the computational issues, Fig. 4 shows the behavior of versus by considering the use of non-optimized codes running on a single-core desktop PC at 1.8 GHz. Of course, the computational time required by the ADS-CP is higher than that of the ADS method whatever vs. the array size (e.g., when —Fig. 4). This is due to the “CP” step within the hybrid procedure. A significant speed enhancement is expected whether ad-hoc algorithms (out-of-the-scope of the present communication) for the convex minimization are used instead of the ‘fmincon’ subroutine. The next experiment is aimed at preliminarily comparing the ADS-CP with state-of-the-art approaches for the design of non-uniformly spaced and tapered layouts. Towards this end, an array with , and dB, has been synthesized starting from a (52, 26, 12, , and it has 13)-ADS [17] and enforcing


Fig. 6. Equi-Ripple Sidelobes . Power pattern of the (a) and amplitudes of the corresponding optimal ADS-CP layout layout (b) versus the state-of-the-art BCS solution [6].

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Fig. 8. Shaped Sidelobes . Power pattern of the optimal (a) and amplitudes and phases of the corresponding ADS-CP layout layout (b).

are known to be a potential problem for this type of arrays, a set of additional experiments has been carried out in order to address the “beam steering” case by assuming (9) arrangement has been synthesized by endB [Fig. 7]. The result obtained by choosing the from [17] indicates that the ADS-CP layout enables the synADS thesis of thinned arrays not exhibiting any grating lobes until [Fig. 7]. Such lobes are actually unavoidable because of the underlying lattice, which yields a periodicity in with period [13]. This result shows that steered arrays without grating lobes can be synthesized by the proposed methodology by simply enforcing the de(Fig. 7). The steered array weights can be sired mask in . then computed by introducing the known linear phase term into More specifically, a and forcing

Fig. 7. Equi-Ripple Sidelobes optimal ADS-CP layout

. Power pattern of the .

been compared with a Bayesian compressive sensing (BCS) sparse layout4 with the same aperture, mainlobe size, and number of elements, provided in Fig. 5(d) of [6]. As expected, the BCS strategy achieves dB lower sidelobe levels [ dB vs. —Fig. 6(a)] and amplitude ratios [ dB—Fig. 6(b)] because of the increased DoFs in the synthesis process [6]. However, the ADS-CP guarantees an enhanced modularity of the final layout thanks to the regularity of the underlying lattice [which, on the contrary, is not matched by the BCS technique—Fig. 6(b)]. It is worth remarking that all the above results have been obtained assuming broadside steered patterns with mask constraints enforced ). Since grating lobes only within the visible range (i.e., 4The two strategies slightly differ in terms of formulation, since the BCS considers a pattern matching problem with no lattice constraints [6], while the ADS-CP solves a mask-constrained synthesis with a lattice-compliant layout. Nevertheless, BCS is worth as a term of comparison because of its effectiveness with respect to state-of-the-art non-uniform array design methods [6].

B. Shaped Sidelobes The last example is aimed at assessing the flexibility of the proposed hybrid approach when dealing with more complex pattern masks. Towards this end, the following mask (10) has been enforced considering the assumptions of Fig. 3. The plot of the synthesized optimal pattern [Fig. 8(a)] confirms the ability of the ADS-CP to match arbitrary (non-symmetric) far-field constraints [Fig. 8(a)]. Moreover, the corresponding array layout [Fig. 8(b)] dB—Fig. 8(b)] although no presents a reduced constraints on this parameter have been imposed during the synthesis process.

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IV. CONCLUSION In this communication, a hybrid ADS-CP technique for the design of non-regular linear arrays complying with arbitrary power pattern upper bounds is described. The procedure comprises a fully-analytical ADS-based thinning step and a successive CP solver to compute the element excitations. From the methodological viewpoint, the main advances of this work over the state-of-the-art include (i) the hybridization of a fully-analytical thinning strategy with a local optimization tool to enable the design of wide aperture light arrays with fully controlled sidelobes with affordable computational costs, and (ii) the generalization of the ADS thinning strategy [13] to deal with user-defined pattern constraints. Moreover, the numerical analysis has pointed out the following isthe ADS-CP technique is able to produce massively-thinned sues: arrangements over a user-defined lattice [e.g., Fig. 2(b)] and to effithe ciently control their radiation features [e.g., Figs. 2(a) and 8(a)]; cyclic-shift search yields several trade-off layouts from a single ADS sequence [Figs. 1(c) and 3] depending on the user-defined optimality the sidelobe control of the ADS-CP is significantly better, criterion; as expected, than that of the ADS method. Such an enhancement is more unlike the ADS strategy, evident as the array size increases (Fig. 4); the synthesis time turns out to be non-negligible especially when large apertures are at hand (Fig. 4) due to the CP subroutine. Future works will be devoted to extend the ADS-CP method to planar and conformal layouts also taking into account more realistic models for the array elements. More in detail, the introduction of suitable techniques to account for mutual coupling among non-equispaced elements, which are out of the scope of the present communication, and the analysis of the arising pattern performance are currently under investigation and will be addressed in future communications. Moreover, the integration of efficient solvers for large CP problems could be of interest from the computational viewpoint. ACKNOWLEDGMENT The authors wish to thank Prof. T. Isernia for useful discussions and comments on the CP formulation and the hybrid ADS-CP. Moreover, they would like to thank the anonymous reviewers for their constructive comments and suggestions aimed at improving the manuscript.

REFERENCES [1] P. V. Brennan, A. Narayanan, and R. Benjamin, “Grating lobe control in randomised, sparsely populated MIMO radar arrays,” IET Radar Sonar Navig., vol. 6, no. 7, pp. 587–594, Aug. 2012. [2] X. Zhuge and A. G. Yarovoy, “Sparse multiple-input multiple-output arrays for high-resolution near-field ultra-wideband imaging,” IET Microw. Antennas Propag., vol. 5, no. 13, pp. 1552–1562, Oct. 2011. [3] C. Luison et al., “Aperiodic arrays for spaceborne SAR applications,” IEEE Trans. Antennas Propag., vol. 60, no. 5, pp. 2285–2294, May 2012. [4] R. L. Haupt, “Thinned arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 42, no. 7, pp. 993–999, Jul. 1994. [5] T. Isernia et al., “A hybrid approach for the optimal synthesis of pencil beams through array antennas,” IEEE Trans. Antennas Propag., vol. 52, no. 11, pp. 2912–2918, Nov. 2004. [6] G. Oliveri and A. Massa, “Bayesian compressive sampling for pattern synthesis with maximally sparse non-uniform linear arrays,” IEEE Trans. Antennas Propag., vol. 59, no. 2, pp. 467–481, Feb. 2011. [7] G. Oliveri, M. Carlin, and A. Massa, “Complex-weight sparse linear array synthesis by Bayesian compressive sampling,” IEEE Trans. Antennas Propag., vol. 60, no. 5, pp. 2309–2326, May 2012. [8] V. Murino, A. Trucco, and A. Tesei, “Beam pattern formulation and analysis for wide-band beamforming systems using sparse arrays,” Signal Process., vol. 56, no. 2, pp. 177–183, Jan. 1997. [9] M. Crocco and A. Trucco, “Stochastic and analytic optimization of sparse aperiodic arrays and broadband beamformers with robust superdirective patterns,” IEEE Trans. Audio, Speech, Language Process., vol. 20, no. 9, pp. 2433–2447, Nov. 2012.

[10] A. Trucco and V. Murino, “Stochastic optimization of linear sparse arrays,” IEEE J. Ocean Eng., vol. 24, no. 3, pp. 291–299, Jul. 1999. [11] M. Skolnik, G. Nemhauser, and J. Sherman, “Dynamic programming applied to unequally spaced arrays,” IEEE Trans. Antennas Propag., vol. 12, no. 1, pp. 35–43, Jan. 1964. [12] Y. T. Lo, “A mathematical theory of antenna arrays with randomly spaced elements,” IEEE Trans. Antennas Propag., vol. 12, no. 3, pp. 257–268, May 1964. [13] G. Oliveri, M. Donelli, and A. Massa, “Linear array thinning exploiting almost difference sets,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 3800–3812, Dec. 2009. [14] G. Oliveri and A. Massa, “Genetic algorithm (GA)-enhanced almost difference set (ADS)-based approach for array thinning,” IET Microw. Antennas Propag., vol. 5, no. 3, pp. 305–315, Feb. 2011. [15] T. Isernia, P. Di Iorio, and F. Soldovieri, “An effective approach for the optimal focusing of array fields subject to arbitrary upper bounds,” IEEE Trans. Antennas Propag., vol. 48, no. 12, pp. 1837–1847, Dec. 2000. [16] L. Manica et al., “Design of subarrayed linear and planar array antennas with SLL control based on an excitation matching approach,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1684–1691, Jun. 2009. [17] ELEDIA Almost Difference Set Repository [Online]. Available: http:// www.ing.unitn.it/~eledia/html/ [18] G. Oliveri, M. Donelli, and A. Massa, “Genetically-designed arbitrary length almost difference sets,” Electron. Lett., vol. 45, no. 23, pp. 1182–1183, Nov. 2009. [19] T. Isernia and G. Panariello, “Optimal focusing of scalar fields subject to arbitrary upper bounds,” Electron. Lett., vol. 34, pp. 162–164, 1998. [20] M. D’Urso and T. Isernia, “Solving some array synthesis problems by means of an effective hybrid approach,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 750–759, Mar. 2007.

Weakly Convex Discontinuity Adaptive Regularization for Microwave Imaging Funing Bai, Aleksandra Pižurica, Ann Franchois, Sam Van Loocke, Daniël De Zutter, and Wilfried Philips Abstract—Reconstruction of inhomogeneous dielectric objects from microwave scattering is a nonlinear and ill-posed inverse problem. In this communication, we develop a new class of weakly convex discontinuity adaptive (WCDA) models as a regularization for quantitative microwave tomography. We show that this class includes the Huber regularizer and we show how to combine these methods with electromagnetic solvers operating on the complex permittivity profile. 2D reconstructions of objects from the Institute Fresnel database and experimental data at a single frequency demonstrate the effectiveness of the proposed regularization even when employing far less transmitters and receivers than available in the database. Index Terms—Discontinuity adaptive regularization, electromagnetic scattering, inverse problem, microwave imaging.

I. INTRODUCTION Quantitative microwave imaging [1] forms images of internal sections of objects in a noninvasive and nondestructive way. The images are obtained by processing the scattered field data after illuminating the objects with known incident fields. Earlier regularized iterative methods to solve this nonlinear and ill-posed inverse problem Manuscript received June 25, 2012; revised July 11, 2013; accepted September 22, 2013. Date of publication September 26, 2013; date of current version November 25, 2013. F. Bai, A. Pižurica, and W. Philips are with the Department of Telecommunications and Information Processing (IPI-TELIN-iMinds), Ghent University, B-9000 Gent, Belgium (e-mail: [email protected]). A. Franchois, S. Van Loocke, and D. De Zutter are with the Department of Information Technology (INTEC), Ghent University. Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2013.2283603

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